The present invention relates to spectral analysis, and more particularly relates to a method for robust spectral analysis, suitable for both light-tailed and heavy-tailed data, a spectral analyzer, signal detector and receiver system, which operates in accordance with the novel spectral analysis method.
The spectral density or power spectral density of a signal describes how the energy or variance of a signal or time series is distributed with frequency. Spectral analysis is a widely used method for analyzing the serial and mutual dependence of one-dimensional and multi-dimensional time series data, and has important applications in science and engineering. For example, spectral analysis is applied in signal and image analyses, target detection techniques, pattern recognition and frequency estimation analyses. If f(t) is a finite-energy (square integrable) signal, the spectral density (Φ(ω)) of the signal is the square of the magnitude of the continuous Fourier Transform of the signal, where energy is the integral of the square of the signal, which is the same as physical energy if the signal is a voltage applied to a 1 ohm load.
A periodogram is an estimate of the true spectral density of a signal. Conventionally, a periodogram is computed from a finite-length digital sequence using the fast Fourier transform (FFT). Conventional techniques of spectral analyses, including periodograms and autoregressive models, are known to lack robustness for heavy-tailed data, resulting in poor performance in situations such as outlier contamination. As used herein, heavy-tailed and light-tailed distributions are terms that are used to denote distributions that are relative to Gaussian distributions, wherein the heavy tailed data have outliers in the data. For example, uniform distributions have lighter tails (less outliers) than Gaussian distributions; Student's T distributions and Laplace distributions are heavy-tailed distributions, wherein Cauchy distributions may be described as very heavy-tailed distributions.
Periodograms provide an estimate of the true spectral density of a signal. Periodograms are often computed from finite-length digital sequences using Fast Fourier transforms (FFTs). Typically, raw periodograms are not good spectral estimates because of statistical variability in view of the fact that the variance at a given frequency does not decrease as the number of samples used in the computation increases. This problem may be reduced by using spectral smoothing, and data tapering techniques.
What would be desirable in the field of spectral analysis, and in particular, spectral analyses for heavy tailed data and for data corrupted by heavy-tailed noise, is a method for providing a robust spectral analysis for time series signals including both light-tailed and heavy-tailed data, a signal detector and a receiver system that operate in accordance with the method.